Question

Show that sin 19 degrees +sin 41 degrees + sin 83 degrees = sin 23 degrees + sin 37 degrees + sin 79 degrees

Answer 1

RHS
sin23 + sin 37 + sin 79
=2 sin (37 +23) /2 x cos (37- 23)/2 + sin79
=2 sin 60/2 x cos 14/2 + sin (90-11)
=2 sin30 x cos 7 + cos 11
= 2 x 1/2 x cos (90 -83) + cos 11 
= sin 83 + cos 11

LHS
sin 19 + sin41 + sin83 
= sin (19 +41) + sin 83
= 2 sin (14+19) /2 x cos (41-19) /2 + sin 83
= 2 sin 60/2 x cos 22/2 + sin 83
=2 sin30 x cos 11 + sin 83
= 2 x 1/2 x cos 11 + sin 83 
= cos 11 + sin 83

Answer 2

Answer:

sin 83° + cos 11° = sin 83° + cos 11°

Step-by-step explanation:

LHS,

= sin 41° + sin 19° + sin 83°

= 2 sin 60°/2 × cos 22°/2 + sin 83° [since, sin c + sin d = 2 sin (c+d)/2 cos (c-d)/2]

= 2 × 1/2 cos 11° + sin 83°

= cos 11° + sin 83°

RHS,

= sin 37° + sin 23° + sin79°

= 2 sin 37°/2 × cos 14°/2 + sin 79° [since, sin c + sin d = 2 sin (c+d)/2 cos (c-d)/2]

= 2 × 1/2 × cos 7° + sin (90° - 11°)

= cos (90°-83°) + cos 11°

= sin 83° + cos 11°

Therefore, LHS = RHS

Hence , (proved)